You want to know specifically where "Say Cheese" came from, yes? The answers so far are just reiterating your second statement, that you know it's go get in smiling position. Why didn't it become "Say Whiskey" for example? "Say Tree", etc...? Maybe this is better suited for the English SE?
– BruceWayneApr 17 '17 at 21:59

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For what it’s worth, some photographers have more success using “say sex” - most subjects find it hard to suppress at leas a smile. Personally, I use “bikini” as it combines the two.
– ManngoApr 18 '17 at 3:07

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@BruceWayne - Here in Ecuador, it is "Say Whiskey" actually. So it is cultural.
– ItaiApr 18 '17 at 6:08

In Germany some people use "Spaghettiiiii". Whatever, the less sexually driven versions are more useful for a mixed crowd including children, where you don't want to upset smaller children (and/or their parents) and don't want the teenagers to break down giggling.
– skymningenApr 18 '17 at 10:53

I have heard it said that a better word might be "sh*t", because the sh sound shows the front teeth better. Yet many people won't instruct groups (often groups of mixed background and generation) to say this in order to take a photo. Also, it's a relatively short sound, so the 'tog would have to stay on the ball.
– AJFaradayApr 18 '17 at 10:55

5 Answers
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In France they don't say "cheese", but rather "ouistiti", which is the French for a kind of monkey I believe. These prompts are simply a way to get people's mouths into a "smiling position". How "cheese" or any other word was specifically arrived upon is probably lost to history.

Wikipedia has an article listing the equivalent of "say cheese" in different cultures.

I heard a story about a woman who told her French auntie how to get her photo subjects to smile. But when the lady said "dites 'fromage'!", the results were unsatisying....
– Shawn V. WilsonApr 17 '17 at 20:33

I believe the use of "cheese" in the phrase is because of the position it puts the subject's mouth in. Mouth the word "cheese" slowly and you will notice that the ending "e" or "uh" sound puts the corners of your mouth in a smile.

Other answers here have focused on the why, so I have endeavoured to find a literal answer to the titular question where. According to the OED, the earliest mention of the practice is in Notes & Queries n:o 158 (Series 13, volume 2), which was published in the '30s. Unfortunately, that volume still has a few years before it is released into the public domain, but here is the passage as quoted in the OED:

Another slang use of the word ‘cheese’ was in vogue at Rugby School (…) This was with the meaning ‘smile’ both verb and noun.

As you can see, the term is spoken of in reminiscence, which would suggest it is older than the '30s, but not older than by a generation.

In slovak, they use the word Syr which stands for cheese but pronounced Seer (the word standing for the same concept of cheese is only a coincidence as far as this answer is concerned). As @WilliamAnderson points out in his answer - it's to get the subject(s) to express something of a smile while saying the word.